how does the drop-height of a ball affect the number of times it will bounce?

Answer 1

In theory, any perfectly elastic ball dropped on a perfectly hard flat surface will bounce endlessly. If, instead, there's a slight inelasticity, so that a fraction of the ball's kinetic energy is lost through heat, then there is a geometric decay of the height of successive bounces. The last piece of the puzzle is that after the bounces diminishes to a certain small height, it quits bouncing and just rolls around. So, having a greater drop-height will result in a greater number of bounces before the geometrically diminishing height of bounces reaches that lower threshold height.

Answer 2

No. The height of the bounce is directly related to how high it was dropped. It gets smaller and smaller at a rate constant no matter what height.